IOP PUBLISHING Phys. Med. Biol. 52 (2007) R1–R13
PHYSICS IN MEDICINE AND BIOLOGY
doi:10.1088/0031-9155/52/6/R01
TOPICAL REVIEW
Approximate and exact cone-beam reconstruction with standard and non-standard spiral scanning
Ge Wang1,2, Yangbo Ye3 and Hengyong Yu1
1 Biomedical Imaging Division, VT-WFU School of Biomedical Engineering, Virginia Tech, Blacksburg, VA 24061, USA 2 Biomedical Imaging Division, VT-WFU School of Biomedical Engineering, Wake Forest University, Winston-Salem, NC 27157, USA 3 Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
E-mail: wangg@https://www.doczj.com/doc/a04878428.html,, yangbo-ye@https://www.doczj.com/doc/a04878428.html, and hengyongyu@https://www.doczj.com/doc/a04878428.html,
Received 1 October 2006, in nal form 30 November 2006 Published 19 February 2007 Online at https://www.doczj.com/doc/a04878428.html,/PMB/52/R1 Abstract The long object problem is practically important and theoretically challenging. To solve the long object problem, spiral cone-beam CT was rst proposed in 1991, and has been extensively studied since then. As a main feature of the next generation medical CT, spiral cone-beam CT has been greatly improved over the past several years, especially in terms of exact image reconstruction methods. Now, it is well established that volumetric images can be exactly and efciently reconstructed from longitudinally truncated data collected along a rather general scanning trajectory. Here we present an overview of some key results in this area.
1. Introduction In x-ray computed tomography (CT), projections are typically collected from a source trajectory around an object to be reconstructed. In the 2D case, the fan-beam geometry is the most common, which is dened by a point source and a linear detector array. In the 3D case, cone-beam geometry becomes more and more popular with use of an area detector array. As compared to fan-beam geometry, cone-beam geometry allows 2D data acquisition, large volume coverage and efcient photon utilization. When the object is spherical, cone-beam projections can often be collected from various orientations without any data truncation. However, when the object is rather long, cone-beam projections are longitudinally truncated in many applications due to limitations in detector extent and radiation dose. This long object problem is practically very important. In material science and engineering, specimens are often rod shaped. In medical imaging, patients are quite long. In small animal imaging, mice and rats are fairly similar to the human in terms of anatomical proportions.
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To solve the long object problem at high temporal resolution, fan-beam spiral CT appeared for the rst time in the patent literature in 1986 (Mori 1986). Related pioneering work began in Japan in the late 1980s. Fan-beam spiral CT research assumed a great momentum in the early 1990s (Bresler and Skrabacz 1989, Kalender et al 1990, Crawford and King 1990, Crawford 1991, Polacin et al 1992, Crawford and King 1993, Wang and Vannier 1994). Fan-beam spiral CT was introduced for better temporal resolution than conventional incremental CT, which works in a stepping and shooting mode. As a result, spiral CT produces inconsistent projections on any cross-section, and broadens the slice sensitivity prole (SSP) relative to the counterpart of incremental CT. It appeared at the very beginning that temporal resolution of fan-beam spiral CT was improved at the cost of degraded longitudinal spatial resolution. Actually, spiral CT possesses a very uniform longitudinal sampling pattern and allows retrospective reconstruction, which means that raw data are collected rst, and slices can be retrospectively reconstructed at very small reconstruction intervals. It was proven using a linear system approach that fan-beam spiral CT in an overlapping reconstruction mode possesses a wider bandwidth and thus better longitudinal resolution than incremental CT for a given x-ray radiation dose (Wang and Vannier 1994). It has been recommended that 3–5 slices be reconstructed per patient table increment. Experiments have also demonstrated superiority in longitudinal resolution of fan-beam spiral CT (Kalender et al 1994, Kalender 1995). A further analysis demonstrated that given both an x-ray dose and a longitudinal bandwidth, fan-beam spiral CT actually produces less image noise on average than incremental CT (Wang and Vannier 1997). In other words, fan-beam spiral CT is inherently superior to incremental CT in terms of spatial, contrast and temporal resolution simultaneously. 2. Original Idea The original cone-beam tomography assumes no data truncation at all. Kirillov derived a formula for reconstruction of a complex-valued n-dimensional function from complex-valued cone-beam projections (Kirillov 1961). Smith made the rst attempt to translate Kirillov’s nding into the real space (Smith 1983). Then, an important cone-beam tomography formula was developed under the condition that almost every hyper-plane through a compact function support meets a source locus transversely (Tuy 1983). Independent milestone results on cone-beam tomography were also achieved by Smith and Grangeat, respectively (Smith 1985, Grangeat 1991). Various algorithms were developed for exact image reconstruction from complete cone-beam projections according to Smith’s theory (Smith and Chen 1992), Grangeat’s relationship (Weng et al 1993, Defrise and Clack 1994, Kudo and Saito 1994, Axelsson and Danielsson 1994, Hu 1996), and Tuy’s formula (Zeng et al 1994). While the circular cone-beam scanning is the most widely used to reconstruct a spherical object (Feldkamp et al 1984), the initial efforts were made in 1991 to solve the long object problem with spiral cone-beam scanning by Wang et al (Wang et al 1991, 1993, Kudo and Saito 1991) (gure 1). Since then, spiral cone-beam CT has gradually become a main area of CT research and development. In reference to Bushberg et al (2001) we may characterize the CT development in terms of the seven generations: parallel beam, narrow fan beam, wide fan beam with a rotating data acquisition system, wide fan beam with a rotating source only, electron-beam scanning, spiral fan-beam scanning, and spiral cone-beam scanning with circular multi-slice/cone-beam scanning as a special case. Especially, in medical CT we are now in a rapid transition from multi-slice to cone-beam CT, which highlights the clinical importance of spiral cone-beam scanning. Since the generalized Feldkamp algorithm for approximate spiral cone-beam reconstruction was proposed in 1991, great efforts have been made to derive an exact solution.
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Figure 1. Concept of the spiral cone-beam CT (https://www.doczj.com/doc/a04878428.html,/CTL/).
As pointed out by Wang et al (1997), ‘Cone-beam spiral CT seems an ideal imaging mode. It is desirable and possible that an exact cone-beam reconstruction algorithm be designed that takes longitudinally truncated cone-beam data and is computationally efcient.’ Such an exact algorithm would be particularly critical when the cone angle is sufciently large and approximate cone-beam reconstruction algorithms can no longer performs well. Theoretically speaking, the goal towards an exact spiral cone-beam CT solution is extremely challenging. First, all the original cone-beam-based inversion theories assume that an object to be reconstructed is completely contained in any cone beam. This assumption is unavoidably violated in the context of a long object reconstruction. Second, the combination of the data truncation and the beam divergence makes it nontrivial to merge truncated data into what is required in the original framework. Third, it appears that an exact reconstruction at any point in the long object would need cone-beam data from the whole object based on the inverse Radon transform. In other words, use of data from multiple helical turns appears necessary for exact reconstruction, which would compromise temporal resolution and increase radiation dose as compared to the approximate reconstruction. As a result, the goal of exact spiral cone-beam reconstruction had not been achieved until Katsevich’s ground breaking work in the early 2000s (Katsevich 2002, 2003, 2004b). 3. Developmental stages In retrospect, the development of spiral cone-beam CT may be divided into the following four stages: (1) approximate reconstruction (1991–present), (2) exact reconstruction with multiple turns (1995–2001), (3) exact reconstruction based on PI (π ) lines (2002–2004) and (4) general exact reconstruction (2004–present). 3.1. Approximate reconstruction In the approximate reconstruction stage (1991–present), a number of very successful algorithms were developed using the Feldkamp approach (Feldkamp et al 1984, Wang et al 1991, Kudo and Saito 1991, Yan and Leahy 1992, Smith and Chen 1992, Noo et al 1999, Kachelriess et al 2000, Tang and Hsieh 2004, Tang et al 2006a, 2006b). The key idea is to correct cone-beam data into fan-beam counterparts in a heuristic way. As a primary example, a cone-beam datum along an oblique ray can be approximately converted to the fan-beam
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counterpart along a transverse ray by multiplying the former with the cosine of the angle between the oblique and transverse rays. These algorithms can perform approximate conebeam reconstruction very effectively, especially in cases of incomplete source scanning. The approximate reconstruction is usually associated with high computational efciency and other image quality benets, but it may produce some image artefacts. The larger the cone angle becomes, the worse the approximate algorithms perform in general. Despite options for exact cone-beam reconstruction, approximate cone-beam algorithms remain important. While the Feldkamp cone-beam image reconstruction algorithm addresses circular scanning and spherical specimen reconstruction (Feldkamp et al 1984), we generalized the Feldkamp algorithm to handle exible scanning loci including helical/helical-like scanning trajectories, and reconstruct spherical, rod-shaped and planar specimens (Wang et al 1991, 1992, 1993, 1994). Using the Feldkamp approach, we also developed algorithms for halfscan-based reconstruction (Wang et al 1994), reconstruction with transversely truncated data (Wang 2002, Liu et al 2003), and multiple source-based reconstruction (Liu et al 2001). Other early approximate cone-beam algorithms are also in the ltered backprojection format. For example, Larson et al proposed a nutating slice helical cone-beam CT method (Larson et al 1998), in which appropriate fan-beam projection datasets are obtained to reconstruct a series of slices of equal tilt angles but changing rotation angles (the normal of successive slices denes a nutation and precession with respect to the table motion direction). All these approximate algorithms differ in the scheme of data rebinning (Noo et al 1999, Kachelriess et al 2000), selection of the ltering direction, denition of the reconstruction plane (Feldkamp et al 1984, Larson et al 1998), assignment of weightings (Tang et al 2006a, 2006b), and so on. The newer approximate cone-beam algorithms were developed in reference to the exact cone-beam reconstruction algorithms (Tang and Hsieh 2004). It is emphasized that the merits of the approximate cone-beam algorithms are not only in their computational efciency but also in several aspects of image quality and radiation dose. Given a small or moderate cone angle like what is now used for medical CT, approximate conebeam algorithms may be competitive against exact alternatives in terms of image resolution, image noise, temporal consistency and dose efciency. On the other hand, when the cone angle becomes too large, the optimality of helical cone-beam scanning will be severely compromised near the two ends of the source trajectory. In this case, we may just want to use circular or other types of scanning curves instead. Therefore, it remains the current practice that all the major CT manufacturers still use approximate cone-beam algorithms as the working house. It is interesting to see whether or not the exact cone-beam algorithms will eventually dominate over the approximate cone-beam algorithms in medical imaging applications. 3.2. Exact reconstruction with multiple turns In the stage of exact reconstruction with multiple turns (1995–2001), the exact reconstruction is achieved using the Radon inversion approach. The key idea is to recover the Radon transform from truncated data segments. The essential concepts on the PI line and the minimum detection window were proposed by Tam (1995). As illustrated in gure 2, it is an elegant geometrical property that given any point in a cylindrical object support there is one and only one PI line that passes through the point and intersects twice a standard helical scanning trajectory outside the object support (Danielsson et al 1997, Kudo et al 1998). The minimum detection window Tam dened for any source position is then delimited by the projections of the upper and lower helical turns upon the detector plate. It is clear that although any cone beam is unavoidably truncated in the long object problem setting, any plane through the object support is actually seamlessly covered by multiple truncated fan beams that are measured within the Tam window.
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Figure 2. Illustration of the PI line and minimum detection window.
Then, the Grangeat relationship was extended from the case of a single fan-beam coverage on any plane through an object to be reconstructed to the case of an equivalent coverage with multiple fan beams in a mosaic pattern (Kudo et al 1998, Tam et al 1998). 3.3. Exact reconstruction based on PI lines In the stage of exact reconstruction based on PI lines (2002–2004), the involvement of conebeam data is limited to neighbouring helical turns for exact image reconstruction. It is emphasized that such a de-correlation between the neighbouring and distant data means a dramatic improvement in temporal resolution and radiation utilization. Also, this paradigm change is revolutionary, departing from the Radon inversion scheme. To our best knowledge, this possibility was rst conceived by Danielsson et al (1997). The heuristics seems to be that with a helical cone-beam scan using the Tam window any point on a PI line is viewed from a complete set of orientations as dened in the Orlov theorem (Orlov 1975a, 1975b). Therefore, the exact reconstruction may be done only from data collected on the scanning arc delimited by the two end points of the PI line segment on the scanning trajectory. It is Katsevich who rst presented and improved such a formula (Katsevich 2002, 2004b). While his pioneering work is theoretically intricate, the computational structure is in the ltered backprojection format, which is highly efcient (Noo et al 2003, Yu and Wang 2004b). Based on Katsevich’s work (Katsevich 2004b), Zou and Pan found a backprojection ltration formula (Zou and Pan 2004b), which allows exact reconstruction from the minimum set of data that may even be transversely truncated to a certain degree. A gap in their proof (Zou and Pan 2004b) was later xed (Zou and Pan 2004a). 3.4. General exact reconstruction In the general exact reconstruction stage (2004–present), various algorithms were developed for exact reconstruction from cone-beam data collected along an arbitrary scanning curve. Note that the generalized Feldkamp algorithm Wang et al developed in 1991 (Wang et al
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1991, 1993) can be applied for not only the standard helical cone-beam reconstruction but also image reconstruction from a quite large family of general scanning trajectories. To our best knowledge, that algorithm is the rst general cone-beam algorithm but it only allows approximate reconstruction. It is recognized that several general schemes exist for cone-beam reconstruction, such as Tuy’s formulation (Tuy 1983), Smith’s theory (Smith 1985), Grangeat’s relationship (Grangeat 1991) and Katsevich’s framework (Katsevich 2003). In our opinion, the following distinction may be made between a scheme and an algorithm. A scheme outlines a general procedure for exact reconstruction but it cannot be directly applied before some key component is specied. On the other hand, an algorithm can be directly implemented for image reconstruction from projections. Specically, there is a non-trivial weighting function to be specied in any of the aforementioned four schemes, which can be highly complicated in some practical cases. Once the weighting function of such a scheme is given for a particular imaging geometry, an algorithm can be then coded step by step, and executed on a computer to process a sinogram and produce a tomographic image. The rst work on general exact cone-beam reconstruction algorithm was published by Palamodov (2004). However, we identied an error in his proof and showed an inconsistence between his formula and the Katsevich formula in the case of standard helical scanning (Yu et al 2006d). Hence, although the Palamodov algorithm produces attractive approximate reconstructions, we are in a disagreement with Palamodov regarding the exactness of his work (Palamodov 2006, Yu et al 2006b). Our group rst proved the general validity of the backprojection ltration (BPF) formula beyond the original case of standard helical cone-beam scanning (Ye et al 2004, 2005, Zhao S Y et al 2004, 2005). The main difference between our result and the original proof by Zou and Pan is that they used Katsevich’s geometric argument based on the shape of the involved helix (Zou and Pan 2004b), while our proof is analytic, and applies to quite general scanning loci. A major implication of the generalized BPF formula is that it can be applied for saddle curve (Yu et al 2005b) and n-PI-window-based reconstruction in the nonstandard spiral scanning cases (Yu et al 2005a). Independent results were also reported by other groups (Zou and Pan 2004a, Zhuang et al 2004, Pack et al 2005, Zou et al 2005). Several general exact ltered backprojection algorithms were also developed by our group and others in the case of a general scanning curve (Katsevich et al 2004, Katsevich 2004a, 2005, Ye and Wang 2005, Pack and Noo 2005, Zhuang et al 2005). While our early results assume the scanning curve is continuous, the latest results removed the constraints to cover discontinuous scanning curves (Zou and Pan 2004a, Pack et al 2005, Zou et al 2005a, Pack and Noo 2005, Zhuang et al 2005, Zhao J et al 2005). 4. Biomedical applications The work on general exact cone-beam reconstruction is not only to satisfy a mathematical curiosity but also to solve biomedical imaging problems. Hence, the results along this direction are exemplary of powerful combinations of practice and theory. In the following, we describe ve examples. As shown in gure 3, the rst example is bolus-chasing CT angiography we proposed (Wang and Vannier 2003, Bai et al 2004, 2006). Intravenous injection of contrast media is required to enhance conspicuity of the vasculature and some other low-contrast features in CT. Synchronization of CT imaging with the propagation of contrast bolus can maximize the signal difference between arteries and background in rst-pass studies. We are developing an adaptive and robust bolus-chasing methodology for spiral cone-beam CT angiography in a wide class of diagnostic applications. This will be achieved by instantaneously reconstructing
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Off-Line (2nd Pass) On-Line (1st Pass)
CT Volume Reconstruction CT Fluoroscopy Image Reconstruction Adaptive Table Motion Control
CT Angiography Image Analysis
ltiSli ce
Mu
Comparison Z Motion
Patient Table
Ultrasound/ECG Measurement , Individualized Parameters
Extended Hammerstein Model/Estimator Parameter Adjustment
Figure 3. Bolus-chasing CT angiography using adaptive control techniques, which combines three high-tech components: CT imaging, physiological modelling and adaptive control, in a fundamental way for the next generation system with major clinical benets (Wang and Vannier 2003, Bai et al 2004, 2006).
CT images, dynamically predicting bolus propagation using a system identication approach, and adaptively varying scanning pitch to scan from the aortic arch to the feet for real-time correction of any signicant mismatch between the bolus peak and the imaging aperture. This synergic combination of imaging, modelling and control naturally leads to a variable pitch cone-beam scanning mode, which requires an extension of the exact algorithms from the standard helical CBCT scanning mode into the variable pitch helical CBCT scanning mode (Katsevich et al 2004, Wang and Ye 2004, Zou et al 2005). The second example is our explorative project on development of electron-beam microCT (Wang and Ye 2004, Wang et al 2004). This project is to reduce the temporal resolution of micro-CT for small animal cardiac studies (Wang and Ye 2004, Wang et al 2004). Currently, a state of the art x-ray micro-CT scanner takes about 20 s to acquire a full dataset, which is too slow to capture the rapidly beating heart of small animals. We are exploring the feasibility of the rst electron-beam micro-CT (EBMCT) prototype for cardiac imaging of the mice and rats. We have nished a top-level design and a preliminary physical analysis for the rst EBMCT scanner (Wang et al 2004). As illustrated in gure 4, in this prototype, the magnetic coils precisely focus and steer the electron beam through the evacuated drift tube. The electron beam strikes the tungsten target, and is spirally scanned over multiple spiral turns with an increasing radius. The area detector supported enclosure acquires the cone-beam data during the nonstandard spiral scan. Note that this design can be modied to allow other non-standard loci, such as saddle curves (Pack et al 2004, Yu et al 2005b). While the above two projects employ continuous cone-beam scanning, our third example, triple-source helical cone-beam CT, requires three cone-beam scanning arcs that are not connected (Zhao J et al 2004). To perform exact cone-beam reconstruction, Zhao et al introduced new concepts of inter-helix PI lines and Z windows, formulated a backprojection ltration formula, and produced excellent simulation results (Zhao J et al 2005, 2006). This
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Generalized Tam window Magnetic focus & deflection coils
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Electron gun
Animal chamber Cone-beam x-ray
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Taper CCD camera
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Figure 4. Electron-beam micro-CT, which is a miniature of the well-known electron-beam CT system, intended for dynamic tomographic studies of small animals, especially cardiac imaging and contrast-enhanced studies (Wang and Ye 2004, Wang et al 2004).
reconstruction formula is actually a special case of the work by Pack et al (2005). Furthermore, the general ltered backprojection formula recently derived by Pack and Noo (2005) can be adapted to solve this triple-source cone-beam reconstruction problem as well. The fourth example is the C-arm CBCT system designed by Chen et al (University of Wisconsin-Madison). The main objective is to improve x-ray interventional procedures in a exible CBCT platform that allows exact and immediate image reconstruction of contrast dynamics to evaluate intervention-induced changes in anatomy and functions. They extended the central slice theorem from the parallel-beam geometry into the divergent-beam geometry (Chen and Leng 2005) to obtain spectral information on a volume to be reconstructed directly from cone-beam data collected along orthogonal-arcs implemented on a at-panel detector C-arm system. For instance, time-resolved data can be taken using orthogonal source rotation or translation. The proposed system seems promising in a number of applications such as post-stroke assessment and various cancer therapies. Finally, non-standard spiral cone-beam scanning is also desirable in cardiac SPECT. In 1992, Gullberg and Zeng proposed a ltered backprojection cone-beam algorithm for this application (Gullberg and Zeng 1992). Their algorithm reconstructs images from a short scan along a noncircular planar orbit with the angular range covering less-attenuated data from cardiac radiopharmaceuticals. This algorithm was designed to minimize the attenuation artefacts, and demonstrated to be advantageous over the full-scan counterpart. Inspired by their work and the latest CBCT results, we hypothesize that exact CBCT algorithms can be developed for SPECT with general scanning trajectories as well, which fundamentally differ from x-ray CBCT algorithms because attenuated projections must be rigorously taken into account in the SPECT case while projections are subject to no attenuation in the x-ray CT case. 5. From approximate to exact reconstruction Although the Feldkamp-type reconstruction is approximate, under certain conditions it does produce exact results. Particularly, the longitudinal integral of a volumetric image
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Chen et al. 2006: LCFBP Zhuang et al. 2005, 2006: Shift-invariantFBP Yang et al. 2006a, 2006b: Saddle curve FBP Zhuang and Chen 2006: SM,DM-lines Zhuang et al. 2004: General BPF Noo et al. 2004: Two step FB Pack et al. 2005: General BPF, open chord King et al. 2006: Motioncontained Pan et al. 2005, Zou et al. 2005b: FB CT Chen 2003a: New framework FB Chen 2003b: Alterative proof Pack and Noo 2005: General FBP, M-line Leng et al. 2005: Half-size detector Katsevich 2004a, 2005: Circle +line, circle+circle Zeng et al. 2006: Mammo-CT Katsevich et al. 2004: VP; Katsevich 2004c, 2006a, 2006b: N-PI window, improved LT Noo et al. 2003: Implementation Wei et al. 2005: General FB Noo et al. 2002:Super short scan FB Yu and Wang 2004b: Implementation Palamodov 2004: General FBP Yu et al. 2006d: Palamodov
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Yu and Wang 2004a: Feldkamptype ROI Yang et al. 2004: CB cover Deng et al. 2006: Katsevich Parallel
Bontus et al. 2005: N-PI window Katsevich 2003: General FBP scheme Katsevich 2002, 2004b: Helical FBP Zou et al. 2005: VP
Zou and Pan 2004b: Helical BPF Zou and Pan 2004c: Helical FBP
Ye and Wang 2005: General FBP Ye et al. 2004, 2005: General BPF
Ye et al. 2006,2007: Pseudo-LT Yu et al. 2005b, 2006c: Saddle curve, local ROI; Yu et al. 2006a, Yu and Wang 2007: Motion Correction; Yu, Zhao and Wang 2005: DSLP Yu et al. 2005a: N-PI window
Defrise et al. 2006: New ROI recon condition Zou and Pan 2004a: Note; Zou et al. 2005a: General FBP/BPF Zhao J et al. 2004, 2005, 2006: Triplesource CBCT
Yu et al. 2006e: Circular ROI
Sidky et al. 2005: Shiftinvariant
Zhao S Y et al. 2004, 2005: Unified framework
Yu and Wang 2006: Exact FB LT
Yu, Ye and Wang 2006: Practical CB LT
Figure 5. Roadmap on the recent development of exact cone-beam CT algorithms, where FB stands for fan-beam, CB for cone-beam, FBP for ltered backprojection, LCFBP for locally compensated FBP, BPF for backprojection ltration, M-line for measured line, SM-line for singly measured line, DM-line for doubly measured line, ROI for region of interest, VP for variable pitch helical CBCT, LT for lambda tomography, and DSLP for the differential Shepp–Logan phantom. The connections are basically based on the citations in the corresponding papers.
reconstructed using the Feldkamp algorithm is exact. We recognized that our generalized Feldkamp reconstruction could be similarly formulated in a rotated reconstruction system after cone-beam data are mapped onto the new imaginary detector plane through the new longitudinal axis (Wang et al 1999). Then, the integral of a reconstructed volumetric image along this longitudinal axis is exact either in the case of no cone-beam truncation or when longitudinally truncated cone-beam data can be assembled in a seamless way to cover any plane through the object support. The exact longitudinal integral is nothing but the 2D parallelbeam projection along the integral direction. Therefore, exact stereo views from cone-beam data can be numerically synthesized. More importantly, exact cone-beam reconstruction can be achieved from a sufciently large number of synthetic parallel-beam projections based on the Orlov theorem (Orlov 1975a, 1975b). This perspective can also lead to the traditional sufcient condition for exact cone-beam reconstruction (Smith 1985, Wang et al 1999). 6. Concluding remarks Since Katsevich’s milestone work on exact and efcient CBCT algorithms in the standard helical scanning case (Katsevich 2002, 2004b), the theory and methods for spiral CBCT have been rapidly advanced. As a summary, gure 5 presents the roadmap on the recent development of exact cone-beam CT algorithms based on or inspired by Katsevich’s ndings. All these schemes/algorithms are in the FBP and BPF formats. Rigorously speaking, a paper is generally inspired by multiple previous papers. The use of the arrows to indicate the connection to the literature can be simplistic or even dissatisfactory. Hence, this roadmap only serves as a guideline and must be read with caution. Now, the state-of-the-art FBP and BPF algorithms can reconstruct images exactly from longitudinally or even transversely truncated data collected along quite general scanning trajectories. Undoubtedly, these results will be instrumental for improvements of the next generation CT and micro-CT scanners. The future
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of this eld seems very bright with many more exciting research opportunities for important biomedical applications. Acknowledgments This work is partially supported by NIH/NIBIB grants (EB002667 and EB004287). The authors are grateful for Drs Carl Crawford, Ming Jiang, Xiangyang Tang, Tiange Zhuang, Jun Zhao and Yu Zou for their constructive comments and valuable advice. References
Axelsson C and Danielsson P E 1994 Danielsson, 3-dimensional reconstruction from cone-beam data in O (N-3 Log-N) time Phys. Med. Biol. 39 477–91 Bai E-W, Wang G and Vannier M W 2004 Systems and methods for adaptive bolus chasing computed tomography (CT) angiography US Provisional Patent Application 60/605, 865 (ling date: 31 August 2004) Bai E-W et al 2006 Study of an adaptive bolus chasing CT angiography J. X-Ray Sci. Technol. 14 27–38 Bontus C, Kohler T and Proksa R 2005 EnPIT: ltered back-projection algorithm for helical CT using an n-Pi acquisition IEEE Trans. Med. Imaging 24 977–86 Bresler Y and Skrabacz C J 1989 Optimum interpolation in helical scan computerized tomography Proc. IEEE Int. Conf. Acoust. Speech and Sig. Proc. pp 1472–5 Bushberg J T et al 2001 The Essential Physics of Medical Imaging 2nd edn (Philadelphia: Lippincott Williams and Wilkins) Chen G H 2003a A new framework of image reconstruction from fan beam projections Med. Phys. 30 1151–1 Chen G H 2003b An alternative derivation of Katsevich’s cone-beam reconstruction formula Med. Phys. 30 3217–26 Chen G H and Leng S 2005 A new data consistency condition for fan-beam projection data Med. Phys. 32 961–7 Chen G H et al 2006 Development and evaluation of an exact fan-beam reconstruction algorithm using an equal weighting scheme via locally compensated ltered backprojection (LCFBP) Med. Phys. 33 475–81 Crawford C 1991 Method for reducing skew image artifacts in helical projection imaging USA Patent No. 5046003 Crawford C and King K F 1993 Method for fan beam helical scanning using rebinning USA Patent No. 5216601 Crawford C R and King K F 1990 Computed-tomography scanning with simultaneous patient translation Med. Phys. 17 967–82 Danielsson P E et al 1997 Towards exact 3D-reconstructin for helical cone-beam scanning of long objects: a new arrangement and a new completeness condition International Meeting on Fully Three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine (Nemacolin, PA) Defrise M and Clack R 1994 A cone-beam reconstruction algorithm using shift-variant ltering and cone-beam backprojection IEEE Trans. Med. Imaging 13 186–95 Defrise M et al 2006 Truncated Hilbert transform and image reconstruction from limited tomographic data Inverse Problems 22 1037–53 Deng J et al 2006 Parallel implementation of the Katsevich algorithm for 3D CT image reconstruction J. Supercomput. 38 35–47 Feldkamp L A, Davis L C and Kress J W 1984 Practical cone-beam algorithm J. Opt. Soc. Am. A 1 612–9 Grangeat P 1991 Mathematical framework of cone beam 3D reconstruction via the rst derivative of the Radon transform Mathematical Methods in Tomography ed G T Herman, A K Louis and F Natterer (Berlin: Springer) pp 66–97 Gullberg G T and Zeng G L 1992 A cone-beam ltered backprojection reconstruction algorithm for cardiac single photon emission computed tomography IEEE Trans. Med. Imaging 11 91–101 Hu H 1996 An improved cone-beam reconstruction algorithm for the circular orbit Scanning 18 572–81 Kachelriess M, Schaller S and Kalender W A 2000 Advanced single-slice rebinning in cone-beam spiral CT Med. Phys. 27 754–72 Kalender W A 1995 Thin-section 3-dimensional spiral CT—is isotropic imaging possible Radiology 197 578–80 Kalender W A, Polacin A and Suss C 1994 A comparison of conventional and spiral CT—an experimental-study on the detection of spherical lesions J. Comput. Assist. Tomogr. 18 167–76 Kalender W A et al 1990 Spiral volumetric CT with single-breath-hold technique, continuous transport, and continuous scanner rotation Radiology 176 181–3 Katsevich A 2002 Theoretically exact ltered backprojection-type inversion algorithm for spiral CT SIAM J. Appl. Math. 62 2012–26
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Topical Review
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Topical Review
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